C₇×#539 #90560

order	448
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#539

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₇×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

218 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,19,20,21,22,23);(1,3,4,7)(2,5,6,8)(10,13)(11,15);(2,6)(5,8)(9,11,12,15)(10,14,13,16);(10,13)(14,16)

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11729640, #11729641, #11729642, …
maximal subgroup of	#11729640, #11729641, #11729642, …
maximal normal subgroup of	#11729640, #11729641, #11729642, …
maximal quotient of	C₇×#2313, C₇×#2318, C₇×#2322, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1322);